3 push-forward -> pull-back

The paper http://arxiv.org/abs/0804.1274 of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy.

First of all they introduce a triangulated $2$-category $Dg(X)$ of derived categorical sheaves on a (derived) scheme $X$. It is based on a the idea that a categorification of the theory of modules on a commutative ring $k$ is given by $k$-linear categories: they argue that dg-categories can be used in order to categorify homological algebra in a similar but better way (better in the sens that the non-dg setting seems to be too rigid to allow push-forwards).

The second step is to use, for a given (derived) scheme $X$, the push-forward pull-back along the projection $LX\to X$. For a categorical sheaf $F$ on $X$ on consider its pull-back $p^*F$, which naturally come equipped with a self-equivalence $u$. The rough idea to see this is to consider the pull-back (a-k-a >transgression) along the evaluation map $S^1\times LX\to X$, and then to observe that categorical sheaves on $S^1\times LX$ are categorical sheaves on $LX$ together with a $\mathbb{Z}$-action.

Finally, they conjecture the existence of an $S^1$-equivariant trace $Tr^{S^1}(u)\in D^{S^1}(LX)$. Its $K_0$ would be a candidate for the (categorified) Chern character of $F$.

### Why does this categorify the Chern character ?

If we do the same construction starting with a sheaf of $X$, then we get in the end an element in $\pi_0(\mathcal O_{LX}^{S^1})=HC_0^{-}(X)$ (while the non-$S^1$-invariant trace takes values in $\pi_0(\mathcal O_{LX})=HH_0(X)$).

One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace.

### Follow-up

A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in a long paper in french.

The paper http://arxiv.org/pdf/0804.1274http://arxiv.org/abs/0804.1274 of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy.

First of all they introduce a triangulated $2$-category $Dg(X)$ of derived categorical sheaves on a (derived) scheme $X$. It is based on a the idea that a categorification of the theory of modules on a commutative ring $k$ is given by $k$-linear categories: they argue that dg-categories can be used in order to categorify homological algebra in a similar but better way (better in the sens that the non-dg setting seems to be too rigid to allow push-forwards).

The second step is to use, for a given (derived) scheme $X$, the push-forward along the projection $LX\to X$. For a categorical sheaf $F$ on $X$ on consider its pull-back $p^*F$, which naturally come equipped with a self-equivalence $u$. The rough idea to see this is to consider the pull-back (a-k-a >transgression) along the evaluation map $S^1\times LX\to X$, and then to observe that categorical sheaves on $S^1\times LX$ are categorical sheaves on $LX$ together with a $\mathbb{Z}$-action.

Finally, they conjecture the existence of an $S^1$-equivariant trace $Tr^{S^1}(u)\in D^{S^1}(LX)$. Its $K_0$ would be a candidate for the (categorified) Chern character of $F$.

### Why does this categorify the Chern character ?

If we do the same construction starting with a sheaf of $X$, then we get in the end an element in $\pi_0(\mathcal O_{LX}^{S^1})=HC_0^{-}(X)$ (while the non-$S^1$-invariant trace takes values in $\pi_0(\mathcal O_{LX})=HH_0(X)$).

One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace.

### Follow-up

A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in a long paper in french: http://arxiv.org/pdf/0903.3292.

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The paper http://arxiv.org/pdf/0804.1274 of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy.

First of all they introduce a triangulated $2$-category $Dg(X)$ of derived categorical sheaves on a (derived) scheme $X$. It is based on a the idea that a categorification of the theory of modules on a commutative ring $k$ is given by $k$-linear categories: they argue that dg-categories can be used in order to categorify homological algebra in a similar but better way (better in the sens that the non-dg setting seems to be too rigid to allow push-forwards).

The second step is to use, for a given (derived) scheme $X$, the push-forward along the projection $LX\to X$. For a categorical sheaf $F$ on $X$ on consider its pull-back $p^*F$, which naturally come equipped with a self-equivalence $u$. The rough idea to see this is to consider the pull-back (a-k-a >transgression) along the evaluation map $S^1\times LX\to X$, and then to observe that categorical sheaves on $S^1\times LX$ are categorical sheaves on $LX$ together with a $\mathbb{Z}$-action.

Finally, they conjecture the existence of an $S^1$-equivariant trace $Tr^{S^1}(u)\in D^{S^1}(LX)$. Its $K_0$ would be a candidate for the (categorified) Chern character of $F$.

### Why does this categorify the Chern character ?

If we do the same construction starting with a sheaf of $X$, then we get in the end an element in $\pi_0(\mathcal O_{LX}^{S^1})=HC_0^{-}(X)$ (while the non-$S^1$-invariant trace takes values in $\pi_0(\mathcal O_{LX})=HH_0(X)$).

One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace.

### Follow-up

A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in a long paper in french: http://arxiv.org/pdf/0903.3292